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NASA Technical Memorandum 4165

Improved Tangent-Cone Method

for the Aerodynamic Preliminary

Analysis System (APAS)

Version of the Hypersonic

Arbitrary-Body Program

Christopher I. Cruz

Langley Research Center

Hampton, Virginia

Gregory J. Sova

Rockwell International Corporation

North American Aircraft Operations

E1 Segundo, California

National Aeronautics andSpace Administration

Office of Management

Scientific and TechnicalInformation Division

1990

Abstract

The Aerodynamic Preliminary Analysis System

(APAS) utilizes a modified version of the HypersonicArbitrary-Body Program (HABP) Mark III code in

its analysis rationale. Four methods are considered

for incorporation into the code as the tangent-conemethod. The combination of second-order slender

body theory and the approximate solution of Ham-

mitt and Murthy shows the best agreement with theexact numerical solutions and is thus included in the

APAS production version of the HABP code.

Introduction

The Aerodynamic Preliminary Analysis System

(APAS, refs. 1 and 2) uses a modified version of

the Hypersonic Arbitrary-Body Program (HABP)

Mark III code (ref. 3) in its analysis rationale. An

integral part of such an analysis is the calculationof inviscid pressure distributions on arbitrary sur-

faces. Vehicle fuselages are often somewhat conical

in shape; thus a method of predicting pressure drag

for such shapes is required.

Four impact pressure methods are evaluated for

their ability to predict the zero-angle-of-attack invis-

cid pressure coefficients of sharp cones with angles of5°, 7.5 °, 10 °, 12.5 °, 15 °, 17.5 °, 20 °, 30 °, 40 °, and 50 °.

These predictions are then compared with the exactsolution for air. Finally, a method is chosen for use

in the APAS production version of the HABP code.

Symbols

K

Mns

Moc

P

poc

qc¢

Voc

5

0c

0,

7

P_c

pressure coefficient, (p - P_c ) / qoc

constant in Newtonian pressure coeffi-

cient equation

Mach number normal to the shock

free-stream Mach number

local pressure, lbf/ft 2

free-stream pressure, lbf/ft 2

dynamic pressure, 1/2p_cV2

free-stream velocity, ft/sec

impact angle, deg

cone half-angle, deg

shock angle, deg

ratio of specific heats

free-stream density, lbm/ft 3

Description of Prediction Methods

The four impact methods evaluated for pres-

sure coefficient prediction were (1) Newtonian theory

(ref. 3), (2) the original HABP Mark III tangent-cone

empirical method (ref. 3), (3) the Edwards tangent-

cone empirical method (ref. 4), and (4) a combinationof second-order slender-body theory and the approx-

imate cone solution of Hammitt and Murthy (ref. 5).

Modified Newtonian theory yields a pressure co-

efficient which is a function only of impact angle:

Cp = K sin 2 6

where K is equal to the stagnation-point pressure

coefficient (ref. 6).Both the HABP Mark III and the Edwards ver-

sions of the tangent-cone empirical method calculate

pressure coefficient as a function of Mach number and

impact angle:

48Mn2s sin 2 6

Cp- 23M2s _ 5

The difference between these two methods lies in the

empirical equations for Mach number normal to theshock. For the HABP Mark III version,

Mns = 1.090909M_c sin 6 + exp(-1.090909Moc sin 6)

For the Edwards version,

Mns = (0.87Moc - 0.544) sin 6 + 0.53

The last method uses a combination of second-

order slender-body theory and the approximate conesolution of Hammitt and Murthy. The pressure

coefficient found with this method is given by

Cp - '/2pocV_ \ \ _ + 1 oc sin2 O, 7./

1 + 7Moc(O_ - Oc) 2cos 20s1 T [-_--- ]i/__-Os } -1)

Results and Discussion

Figures 1 to 10 present inviscid pressure coeffi-

cients (zero angle of attack) for sharp cones with half-angles from 5° to 50 ° and Mach numbers from 1.5

to 25. Each figure contains pressure coefficients cal-culated with each of the four prediction methods as

well as exact values from the tables of Kopal (ref. 7)

and Jones (ref. 8).

For all cone half-angles investigated, Newtonian

theory underpredicts pressure coefficient throughout

the entire Mach number range. Adjustment of the

Newtonian constant from K = 2 to K = 2("/+ 1)x ('_ + 7)/('7 + 3) 2 (ref. 1) would give a reasonable re-

sult for Mach numbers greater than 10, when the in-

viscid pressure coefficient is relatively constant with

respect to Mach number.

The HABP Mark III tangent-cone empirical

method does a better job than Newtonian theory,

but at Mach numbers less than 5 it also greatlyunderpredicts the exact solutions, as shown in

figures 7 to 10.

The Edwards tangent-cone empirical method is avast improvement over the HABP Mark III method.

At smaller cone half-angles, results from the Edwards

method match the exact values closely for Mach

numbers of 1.5 and up. However, as cone half-angleincreases, the discrepancy between the results from

the Edwards method and the exact values grows

larger.

By far, the best of the methods evaluated is that

referred to in figures 1 to 10 as the "2nd Order Slen-

der Body + Hammitt/Murthy" method. With few

exceptions, this method predicts the inviscid pres-

sure coefficient at zero angle of attack with great ac-

curacy. (In most cases, there is less than 1 percentdifference between predictions and the exact values

of Kopal and Jones.) Figures 9 and 10 show the

peculiarities which can occur when this method is

used for large cone half-angles. This degeneration

of calculated pressure coefficient corresponds to the

physical existence of detached shocks for larger cone

half-angles at low supersonic speeds. It is importantto note, however, that even with these discontinu-

ities, this method is still far superior to the otherthree methods considered.

Conclusions

The combination of second-order slender-bodytheory and the approximate cone solution of Ham-

mitt and Murthy is the superior method of those eval-

uated. It is thus included in the Aerodynamic Pre-

liminary Analysis System (APAS) production version

of the Hypersonic Arbitrary-Body Program (HABP)

code as the tangent-cone method. The Newtonian

theory, original HABP Mark III tangent-cone, and

Edwards tangent-cone methods all have applicabilitywithin given restrictions, but outside of these restric-

tions they may yield misleading results.

NASA Langley Research CenterHampton, VA 23665-5225January 4, 1990

References

1. Bonner, E.; Clever, W.; and Dunn, K.: AerodynamicPreliminary Analysis System II. Part I--Theory. NASACR- 165627, 1981.

2. Divan, P.: Aerodynamic Preliminary Analysis System II.Part H--User's Manual. NASA CR-165628, 1981.

3. Gentry, Arvel E.: Hypersonic Arbitrary-Body Aero-dynamic Computer Program (Mark III Version). Vol-ume I--User's Manual. Rep. DAC 61552, Vol. I (AirForce Contract Nos. F33615 67 C 1008 and F33615 67 C

1602), McDonnell Douglas Corp., Apr. 1968. (Availablefrom DTIC as AD 851 811.)

4. Pittman, Jimmy L. (appendix by C. L. W. Edwards):Application of Supersonic Linear Theory and HypersonicImpact Methods to Three Nonslender Hypersonic AirplaneConcepts at Mach Numbers From 1.10 to 2.86. NASATP- 1539, 1979.

5. Hammitt, A. G.; and Murthy, K. R.: ApproximateSolutions for Supersonic Flow Over Wedges and Cones.AFOSR TN 59-304, U.S. Air Force, Apr. 1959. (Avail-able from DTIC as AD 213 088.)

6. Truitt, Robert Wesley: Hypersonic Aerodynamics.Ronald Press Co., c.1959.

7. Staff of Computing Section, Center of Analysis (underthe direction of Zden_k Kopal): Tables of SupersonicFlow Around Cones. Tech. Rep. No. 1 (NOrd ContractNo. 9169), Massachusetts hast. of Technology, 1947.

8. Jones, D. J.: Tables of lnviscid Supersonic Flow AboutCircular Cones at Incidence -'/= 1.4. AGARDograph 137,Pt. I, Nov. 1969.

Cp

.10

.08

.06

.O4

.02

0

2nd Order Slender Body

+ Hammitt/Murthy

........ Edwards Corrected Tangent Cone

Mark III HABP Tangent Cone0......... Newtonian Theory (K = 2)

o Exact (Kopal, ? = 1.405)_0[] Exact (Jones, y = 1.4)

m

i I f l , I , I , I

5 10 15 20 25

Mach no.

Figure 1. Pressure coefficient versus Mach number for 5° sharp cone.

Cp

.12

.10

.08

.06

.04

.02

O

0


+ Hammitt/Murthy


Mark III HABP Tangent Cone

......... Newtonian Theory (K = 2)

0 Exact (Kopal, y = 1.405)

, [] Exact (Jones, _, = 1.4)

..._, El"

L I I I I l I I I I

5 10 15 20 25

Mach no.

Figure 2. Pressure coefficient versus Mach number for 7.5 ° sharp cone.

3

Cp

.2OO

.175

.150

.125

.100

.075

.050

0

0


+ Hammitt/Murthy


0

[]


Newtonian Theory (K = 2)

Exact (Kopal, 7 = 1.405)

Exact (Jones, T = 1.4)

i Jill Ill Ill II iI .ill • _liJ I ill II Jl liliillllQi_ II "I ll'li lillil I I IIil l" III lillil

, t l I , I J I i I

5 10 15 20 25

Mach no.

Figure 3. Pressure coefficient versus Mach number for 10 ° sharp cone.

Cp

.30

.25 _-

.2O

.15

.10


+ Hammitt/Murthy



........ Newtonian Theory (K = 2)

0 Exact (Kopal, T = 1.405)

[] Exact (Jones, _, = 1.4)

.....___ El []

11 li fill .llli111111 .l.lili_ilil. I_I Ill ..... Ill. 111* I i 111 lliill *Jill" IIi

.05 I I , I i I , I ,

0 5 10 15 20

Mach no.

Figure 4. Pressure coefficient versus Mach number for 12.5 ° sharp cone.

I

25

Cp

.5

.4

.3

.2

.1


+ Hammitt/Murthy




o Exact (Kopal, 7 = 1.405)

[] Exact (Jones, 3' = 1.4)

J I I I I I I I I I

0 5 10 15 20 25

Mach no.


Cp

.6

.5

.4

.3

.2

.1


+ Hammitt/Murthy




o Exact (Kopal, _,= 1.405)

[] Exact (Jones, 'y = 1.4)

......................................................... ._-,.-.--._.-- .........

0

J I , I i I J I i I

5 10 15 20 25

Mach no.

Figure 6. Pressure coeffÉcient versus Mach number for 17.5 ° sharp cone.

Cp

.7

.6

.5

.4

.3

.2

0


+ Hammitt/Murthy




O Exact (Kopal, T = 1.405)

[] Exact (Jones, 7 = 1.4)

, I , I , I i I , I

5 10 15 20 25

Mach no.

Figure 7. Pressure coefficient versus Mach number for 20° sharp cone.

Cp

1.2

1.0

.8-

,6 --

.4

0

IiiiiIII

I I I N

iIIII IIII

0

[]


+ Hammitt/Murthy

Edwards Corrected Tangent Cone



Exact (Kopal, _, = 1.405)

Exact (Jones, ? = 1.4)

llllJll .... lllllIllllllNlIilIIlilJllllllliJJllmlJinliJlJllJJlIllllJJiJnl

i I i I i I i I i I

5 10 15 20 25

Mach no.


,

Cp

16

14

1.2 -

10-

8

0


+ Hammitt/Murthy



......... Newton°an Theory (K = 2)

o Exact (Kopal, ",f= 1405)

n Exact (Jones, 7 = 1 4)

J-L_

.... _"....... -r....... 3....... -I....... _ ....... [....... ;....... T ....... 7....... ]

5 10 15 20 25

Mach no


2.00

1.75

Cp 150

1.25


+ Hammitt/Murthy



......... Newton°an Theory (K = 2)

o Exact (Kopal, 7 = 1.405)

_/\_:_,__j rl Exact (Jones, 7 = 1.4)

1.00 , I i I J I n 1 i I

0 5 10 15 20 25

Mach no.


7_

Report Documentation PageNallona' AeronauTIcs an(_

S_a(e Ad nn,p,s1,_lon

1. Report No. / 2. Government Accession No. 3. Recipient's Catalog No.

NASA TM-4165 14. Title and Subtitle

Improved Tangent-Cone Method for the Aerodynamic PreliminaryAnalysis System (APAS) Version of the Hypersonic Arbitrary-Body Program

7. Author(s)

Christopher I. Cruz and Gregory J. Sova

'9. Performing Organization Name and Address

NASA Langley Research CenterHampton, VA 23665-5225 11.

12. Sponsoring Agency Name and Address 13.

National Aeronautics and Space Administration

Washington, DC 20546-0001 14.

5. Report Date

February 1990

6. Performing Organization Code

8. Performing Organization Report No.

L-16404

10. Work Unit No.

506-49-11-01

Contract or Grant No.

Type of Report and Period Covered

Technical Memorandum

Sponsoring Agency Code

15. Supplementary Notes

Christopher I. Cruz: Langley Research Center, Hampton, Virginia.Gregory J. Sova: Rockwell International Corporation, North American Operations, E1 Segundo,California.

16. Abstract

The Aerodynamic Preliminary Analysis System (APAS) utilizes a modified version of theHypersonic Arbitrary-Body Program (HABP) Mark III code in its analysis rationale. Fourmethods are considered for incorporation into the code as the tangent-cone method. Thecombination of second-order slender body theory and the approximate solution of Hammitt andMurthy shows the best agreement with the exact numerical solutions and is thus included in theAPAS production version of the HABP code.

17. Key Words (Suggested by Authors(s))

HypersonicsTangent cone

18. Distribution Statement

Unclassified--Unlimited

Subject Category 02

19. Security Classif. (of this report)unclassified 120"SecurityClassif(°fthispage)Unclassified 121N°'°fPages22"Price8 A02

NASA FORM 1626 OCT 86

For sale by the National Technical hfformation Service, Springfield, Virginia 22161-2171

NASA.Langley, 1990

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